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Inelastic neutron scattering investigations of molecular nanomagnets
Inorganica Chimica Acta 361 (2008) 3771–3776
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Review
Inelastic neutron scattering investigations of molecular nanomagnets G. Amoretti a, R. Caciuffo b,*, S. Carretta a, T. Guidi c, N. Magnani b, P. Santini a a b c
Dipartimento di Fisica, Università di Parma, I-43100 Parma, Italy European Commission, Joint Research Centre, Institute for Transuranium Elements, Postfach 2340, D-76125 Karlsruhe, Germany Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin, Germany
a r t i c l e
i n f o
Article history: Received 29 January 2008 Received in revised form 4 March 2008 Accepted 6 March 2008 Available online 16 March 2008 Dedicated to Dante Gatteschi. PACS: 75.50.Tt 75.10.Jm 75.40.Gb 75.45.+j
a b s t r a c t Inelastic neutron scattering, probing the temporal spin–spin correlation at the microscopic scale, is a powerful technique to study the magnetic behaviour of molecular nanomagnets. Experiments at different energy scales and different energy-transfer resolution allow precise determinations of the parameters defining the effective Hamiltonians used to model the diverse physical properties exhibited by this class of materials. The intrinsic disadvantage of the technique (low flux, requiring a sample mass in the gram scale) is over-compensated by the large amount of information that can be straightforwardly extracted. Zero-field splittings and exchange interactions can be determined with a large degree of confidence, shedding light on important issues such as magnetization tunnelling in giant-spin clusters, or the occurrence of quantum coherence phenomena and their consequences on macroscopic observables. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Inelastic neutron scattering Molecular nanomagnets Spin hamiltonian
Giuseppe Amoretti is Full Professor of experimental physics and Head of the Physics Department of the University of Parma. In 1982/83 he worked two years at the Département de Recherche Fondamentale, Centre d’Etudes Nucléaires de Grenoble, with a fellowship of the European Institute for Transuranium Elements in Karlsruhe, Germany. He was the coordinator of the INFM (D Section) network on ‘‘Fundamental Magnetic Interactions”. He was member of the ‘‘Board of Directors of the European Actinide Research Development Society”. He is the author or coauthor of about 140 publications concerning the physics of condensed matter, in particular the properties of rare-earth, actinide and transition metal compounds. His current activity is mainly focused on molecular magnetism. His contribution is primarily theoretical and is related to the design and the interpretation of experiments performed by neutrons or by other spectroscopic and bulk techniques (EPR, NMR, NQR, lSR, Mössbauer, magnetic measurements, specific heat). Since 1986 he has performed many neutron scattering experiments, at RAL (UK) and ILL (France).
* Corresponding author. Tel.: +49 7247951382; fax: +49 7247951599. E-mail address: [email protected] (R. Caciuffo). 0020-1693/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ica.2008.03.047
Roberto Caciuffo is Head of the Actinide Research Department at the European Commission’s Institute for Transuranium Elements in Karlsruhe, Germany, and Full Professor of experimental physics at the Università Politecnica delle Marche, Ancona, Italy. He has previously been an experimental physicist at the Institut LaueLangevin (France), at the Centre d’Etudes Nucléaires de Grenoble (France), and at the Rutherford Appleton Laboratory (UK). He is a member of advisory boards of several international institutions. His current research is focussed on the study of magnetic correlations and multipolar order in d- and f-electron systems by neutron and synchrotron radiation X-ray scattering. He is author or coauthor of over 200 publications in leading international journals.
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Stefano Carretta obtained both his degrees from University of Parma (Italy), doing his PhD research on molecular magnetism with Prof. G. Amoretti. In 2005 he has been research fellow of the National Institute for the Physics of Matter (INFM) and at the Faculty of Science at Parma. His research has been mainly focused on theoretical modelling of the quantum behaviour of magnetic molecules. The aim of this research is twofold: the understanding of fundamental phenomena like quantum tunnelling and the identification of systems suitable for technological applications, especially in the field of Quantum Computation. He works in close collaborations with experimental groups, and he actively participates to neutron scattering experiments. He has published more than 50 research papers. In 2006 he has been awarded the ”Le Scienze” and President of the Italian Republic medal for his research on molecular nanomagnets.
Tatiana Guidi graduated in physics in 2001 from the University of Camerino, Italy. She obtained her PhD in Material Science Engineering at the Università Politecnica delle Marche, Italy, studying spin excitations and magnetic properties of Molecular Nanomagnets, using neutron scattering technique. She continued her activity there as a post Doc and spent one year as a guest researcher at NIST Center for Neutron Research (USA). She then joined the Junior Research Group ‘‘Magnetism and Superconductivity of Quantum Materials” of Hahn-Meitner-Institut, Berlin as a Post Doc (2006–2008). Since January 2008 she has been appointed as instrument scientist in the excitation group of the ISIS spallation source, Rutherford Appleton Laboratory, UK, where she will continue with her activity in the field of molecular nanomagnetism. She has published more than 20 papers in international peer-reviewed journals.
Nicola Magnani was born in 1976. He graduated in Physics in the year 2000 at the University of Parma (Italy), where he also obtained his PhD in 2003. After post-doc experiences in the field of molecular nanomagnetism and magnetocaloric effect, since 2006 he holds a Research Fellow position at the at the European Commission’s Institute for Transuranium Elements in Karlsruhe (Germany). His research interests mainly concern the fundamental magnetic properties of d- and f-electron systems, both from the theoretical point of view (crystal-field theory) and by several experimental techniques (mainly inelastic neutron scattering). He published more than 50 papers in international peer-reviewed journals.
Paolo Santini got his PhD in Physics at the University of Lausanne, Switzerland (1995), where he continued until 1999 as lecturer and Assistant Professor. From 1999 to 2002 he worked at the Clarendon Laboratory (Oxford), with funds provided by the Swiss National Science Foundation. In 2002 he returned to Italy with a grant funded by the Italian government to reduce the Italian ‘‘brain-drain”. Since 2006 he has been Associate Professor of condensed matter physics at the University of Parma. His research is in theoretical condensed matter physics: strongly correlated electron systems, magnetism in rare-earth and actinide compounds, low-dimensional systems such as spinchains or semiconductor surfaces, and more recently molecular magnets. He has been involved both in fundamental problems and in the investigation of specific materials. For the latter aspect he has established collaborations with several experimental groups, and he regularly visits large-scale facilities to participate in neutron scattering experiments. He is author or coauthor of about 70 publications in leading international journals.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic neutron scattering cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero-field splitting and tunnelling of the magnetization in single molecule magnets . . . . . Antiferromagnetic molecular rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Quantum oscillations of the total spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Tunneling of the Néel vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.
1. Introduction Following the discovery of slow magnetic relaxation and quantum tunneling in Mn12, molecular nanomagnets (MNM) have attracted considerable attention from physicists and chemists alike [1]. Unique applications have been envisaged as ultra-high density memory devices, or for the implementation of quantum computing algorithms exploiting the superposition of single particle quantum states [2–5]. In the absence of reliable ab-initio calculation methods, the physics of MNM is conveniently described by an effective spin Hamiltonian containing all the relevant intra-cluster interactions, that is the Heisenberg exchange, the single-ion zero-field splitting (ZFS), the magnetic dipole–dipole interaction, and the Zeeman term (in presence of an external magnetic field B): H¼
X
J ij si sj þ
i>j¼1;N
þ
X i>j¼1;N
X X i¼1;N
q ^ q ðsi Þ bk ðiÞO k
si Dij sj lB
k
i¼1;N
~ðiÞ si : Bg
ð1Þ
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Here, N is the number of magnetic centres forming the cluster, J ij are ^ q ðsi Þ are Stevens operator equivalents of pair exchange integrals, O k q rank k (k even and 64 for d electrons), bk ðiÞ are ZFS coefficients, ~ ðiÞ are g-factor tenDij is the dipole–dipole interaction tensor, and g sors. The spin Hamiltonian, which phenomenologically describes the magnetic interactions within each molecule, is widely used in molecular magnetism. This approach eliminates all the orbital coordinates of the system and allows to easily take advantage of the molecular symmetry. In particular, the number of non-zero indeq pendent bk ðiÞ parameters is restricted by group theory. Where the Heisenberg term is dominant, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the so-called strong-exchange limit, the modulus SðS þ 1Þ of the total spin P S ¼ i si is practically conserved and the energy spectrum is formed by multiplets of defined S. In this approximation, the anistropic part of (1) can be projected on a given multiplet (usually the ground state), leading to XX q q ^ ðSÞ l B g ~ S; HZFS ¼ Bk O ð2Þ B k
k;q
X
... ... ... ... ... ... ... ...
Bqk
q
with parameters calculated by projection coefficients from the q bk ðiÞ and Dij parameters appearing in Eq. (1).
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The anisotropic part of the spin Hamiltonian (1) can be written in terms of irreducible tensor operators (ITO) of rank k P 1. Thus a mixing of states with different S and M, or at least states with different S if the anisotropy is purely axial, is expected. Difficulties arising from S-mixing can be dealt with the procedure described in [6]. The parameters defining the Hamiltonians (1) and (2) must be considered as phenomenological quantities to be determined from experiment. For this purpose, inelastic neutron scattering (INS) is a very useful technique, in that it gives access to eigenvalues and eigenstates through the energy and the intensity of the resonances appearing in the scattering cross-section. Several applications of INS to the study of MNM have been reported in the literature, often giving information crucial to understand the macroscopic behaviour of the investigated systems [7–19]. In the following, after a brief presentation of the appropriate INS cross-section formulae, a few selected examples showing the potential of the technique will be reviewed. 2. Inelastic neutron scattering cross-section The double-differential magnetic cross-section for unpolarised neutrons is given by [20] X X o2 r kf ^ aQ ^ bÞ ¼ ðcN r 0 Þ2 e2W ðda;b Q g j F j ðQ Þg ‘ F ‘ ðQ Þ expðiQ Rj‘ Þ oXoEf ki a;b j;‘ X pm hm j saj j lihl j sb‘ j midðhx þ Em El Þ; ð3Þ m;l
where a; b are Cartesian coordinates, saj is the a component of the spin operator of the jth ion, j mi is the eigenfunction with energy Em and occupation probability pm . We use the conventional definition of scattering vector Q ¼ ki kf and energy transfer hx ¼ Ei Ef , where EiðfÞ and kiðfÞ are the incident (final) neutron energy and wavevector, respectively; cN is the neutron magnetic dipole moment in nuclear Bohr magnetons, r 0 is the classical electron radius, g j and F j ðQ Þ are the Landé factor and the dipolar form factor of the jth ion, and e2W is the Debye–Waller factor. The intrinsic linewidth of the excitations and the finite instrument resolution are taken into account by replacing the d function in Eq. (3) by an appropriate spectral weight function, usually a Gaussian or a Lorentzian line-shape. For polycrystalline or powder samples, Eq. (3) must be averaged with respect to the possible orientations of the scattering vector Q. An expression of the resulting average adequate for systems with small magnetic anisotropy is given in [21], whereas one valid for uniaxial anisotropy (regardless of its magnitude) and properly accounting for intramolecular interference effects is reported in [22]. A formula of general validity is [9]: o2 r A kf 2W X ¼ e pm Iml ðQ Þdð hx El þ Em Þ; oXoEf N m ki m;l
ð4Þ
where A ¼ 0:29 barn, N m is the number of magnetic ions, and the function Iml (Q) is defined as h i X 2 j0 ðQRj‘ Þ þ C 20 j2 ðQRj‘ Þ ~szj ~sz‘ Iml ðQ Þ ¼ F j ðQ ÞF ‘ ðQ Þ 3 j;‘ 2 1 þ j0 ðQRj‘ Þ C 20 j2 ðQRj‘ Þ ð~sxj ~sx‘ þ ~syj ~sy‘ Þ 3 2 1 2 ~ þ j2 ðQRj‘ Þ½C 2 ðsxj ~sx‘ ~syj ~sy‘ Þ þ C 22 ð~sxj ~sy‘ þ ~syj ~sx‘ Þ 2 o þ j2 ðQRj‘ Þ½C 21 ð~szj ~sx‘ þ ~sxj ~sz‘ Þ þ C 21 ð~szj ~sy‘ þ ~syj ~sz‘ Þ ; ð5Þ where Rj‘ gives the relative position of ions j and ‘,
C 20 ¼ C 22 ¼
2 1 Rj‘z ½3 1; 2 Rj‘ R2j‘x R2j‘y R2j‘
C 22 ¼ 2 C 21 ¼
Rj‘x Rj‘y R2j‘
Rj‘x Rj‘z
C 21 ¼
R2j‘
;
;
Rj‘y Rj‘z R2j‘
;
ð6Þ
;
and ~saj ~sc‘ ¼ hm j saj j lihl j sc‘ j mi
ða; c ¼ x; y; zÞ:
ð7Þ
The INS spectra are therefore composed by a discrete number of peaks, whose intensity is a linear combination of products of geometrical factors (depending on the momentum transfer Q) and matrix elements (between initial and final states) of the cartesian components of the spin operators sj . The Hamiltonian (1) can be used to calculate Em and all of the ~saj . A comparison between the experimental spectra and the calculated counterparts allows then determining the parameters of the microscopic Hamiltonian. A stringent test for the spectroscopic assignment of the observed transitions is provided by the Q dependence of their intensity, which is essentially determined by the geometry of the cluster and the composition of the spin wavefunctions. This dependence can be easily measured with properly calibrated detectors at different scattering angles. Where the strong exchange limit holds, Eq. (4) can be replaced by a much simpler expression involving the matrix elements of the cartesian components of the total spin S [23]: X X o2 r / pm j hm j Sa j lij2 dð hx El þ Em Þ: oXoEf m;l a¼x;y;z
ð8Þ
This is the same cross-section widely used for the interpretation of crystal-field excitations measured for polycrystalline samples, with the total spin operator replacing the total angular momentum. Its validity in the present case is limited to molecules with uniaxial anisotropy (z being the symmetry direction), or when Q ’ 0.
3. Zero-field splitting and tunnelling of the magnetization in single molecule magnets Quantum tunnelling of the magnetization in molecules with a magnetic ground state and a high Ising-like anisotropy barrier has been widely investigated in the last decade [24,25]. The phenomenon is allowed in presence of ZFS terms breaking the axial symmetry. In that case, the total spin component along the anisotropy axis z is time dependent and the spin state oscillates with time between j S; Si and j S; Si. The degeneracy of the two states associated to opposite sides of the anisotropy barrier is removed by tunnelling, and the tunnel splitting D determines the time dependence of the low temperature magnetic relaxation. The value of D can be estimated quite precisely from magnetization measurements, but it is usually too small to be accessible by INS. For instance, D is of the order of 108 meV in the case of the Fe8Br cluster. Nevertheless, INS leads to a determination of the ZFS Hamiltonian, from which the tunnel splitting can be easily calculated by numerical diagonalization. This is true, provided that all the ingredients of the physical problem are correctly taken into account. An instructive example is that of Fe8Br, one of the first giant-spin MNM studied by neutrons [7,26]. The molecule is characterized by an approximate D2 symmetry. Antiferromagnetic interactions
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relaxation time sðBÞ occur in correspondence of level anti-crossing at B ¼ 0:5 and 1 T. This behaviour is correctly reproduced by the ZFS Hamiltonian derived from the INS results, the trigonal term ^ 3 being responsible for anti-crossing at 0.5 T, and O ^ 2 for the O 4 2 anti-crossing at 1 T.
20
15
10
Energy (meV)
Intensity (arb. units)
3
2
4. Antiferromagnetic molecular rings
1
4.1. Quantum oscillations of the total spin 0
5
0 -0.4
-0.2
0
0.2
0.4
Energy Transfer (meV) Fig. 1. INS spectrum of Fe8Br measured at 9.6 K. The well defined peaks correspond to dipole-allowed transitions between the zero-field split components of the S ¼ 10 ground state, schematised in the inset. The smooth line is the theoretical spectrum for the ZFS Hamiltonian in the strong-exchange limit. From Ref. [7].
between Fe ions lead to a S ¼ 10 ground state, approximately described by eight individual spins s ¼ 5=2, six parallel to each other and two antiparallel. The high-resolution INS spectrum of Fe8Br, taken at T ¼ 9:6 K with an energy resolution of 19 leV at the elastic peak, is shown in Fig. 1. The peaks in the cross-section correspond to transitions between components of the S = 10 ground state, split by a ZFS interaction of D2 symmetry. The analysis of the experimental spectrum with Eqs. 2 and 8 gives a precise estimate of the allowed parameters, B02 ¼ 8:4ð1Þ leV, B22 ¼ 4:02ð3Þ leV, 4 4 0 2 B4 ¼ 0:87ð6Þ10 leV, B4 ¼ 0:1ð1Þ10 leV and B44 ¼ 7:4ð6Þ 104 leV. However, when the Hamiltonian (2) is diagonalised with these set of parameters, one obtains a tunnel splitting of 3:8 1012 meV, four orders of magnitude smaller than the value given by macroscopic measurements. This huge discrepancy reflects the inadequacy of the strong-exchange limit. Indeed, the first excited spin state, a S ¼ 9 multiplet, is about 4 meV above the ground state, and some degree of S-mixing is expected. Although small, S-mixing is very effective in enhancing the tunnel splitting, as it opens highly efficient tunnelling channels. When this fact is properly taken into account, a value of D consistent with that deduced from magnetic relaxation measurements can be obtained [27]. In many cases, INS experiments can lead to a determination of the point symmetry of the cluster in the low-temperature phase, with a sensitivity not parallelled by diffraction measurements. An example is given by the study of the Fe4 cluster [28], a molecule formed by three Fe3+ ions lying at the vertices of an equilateral triangle and by one Fe3+ ion at its centre. X-ray diffraction at room temperature suggests a D3 symmetry, with the C 3 axis perpendicular to the plane of the equilateral triangle. The ground state has a total spin S ¼ 5, with the two lowest quasi-degenerate states approximately given by j S ¼ 5; M ¼ 5i. Intramultiplet excitations have been measured by high-resolution INS experiments ^ 2 term, [29]. A good fit of the results is obtained only if a rhombic O 2 not allowed by the D3 symmetry, is introduced. This term is necessary to account for the manifestation of zero-field quantum tunnelling effects in the magnetic relaxation below 0.2 K [28]. In fact, the tunnel splitting D would be zero if the only non-axial ZFS term ^ 3 , since j S ¼ 5; M ¼ 5i are not were the D3 symmetry-allowed O 4 ^ 3 . Conversely, when a magnetic field connected by any power of O 4 is applied, shortcuts for the tunneling process and minima in the
Ring-shaped cyclic clusters formed by coplanar antiferromagnetically coupled ions have attracted considerable attention for the peculiar low-energy spin dynamics they exhibit and the potential applications as qubits or low-temperature magnetocaloric refrigerants [30]. The possibility to observe quantum tunnelling of the Néel vector in these systems has been extensively debated, and the occurrence of quantum oscillations of the total spin length at anti-crossings between different S multiplets has been clearly demonstrated [31]. For an even number of spins, the ground state has S ¼ 0 and the excitation spectrum is characterized by rotational and spin-wave like bands [32,33]. This is the case of the homometallic Cr8 ring, formed by octahedrally coordinated Cr3+ ions with s ¼ 3=2. Heterometallic rings with S 6¼ 0 can be obtained from a S ¼ 0 homonuclear ring by chemical substitution of one or two magnetic centres [34,35]. Such a substitution affects also the spin topology and the overall macroscopic behaviour of the system [9]. For instance, substitution of Cr3+ with Ni2+, gives a cluster with spin ground state S ¼ 1=2, which could be a suitable candidate for the physical implementation of qubits [5]. For a ring-shaped molecule containing N 1 magnetic centers T and one magnetic centre T 0 , the spin Hamiltonian (1) assumes the form H¼
N 2 X
J TT si siþ1 þ J TT 0 ðsN1 sN þ sN s1 Þ þ
i¼1
þ
N X i¼1
N X
di ½s2z;i
1 si ðsi þ 1Þ 3
gðiÞB si :
ð9Þ
i¼1
ei ½s2x;i
s2y;i
þ
X i>j¼1;N
si Dij sj lB
X i¼1;N
In the above equation it is assumed that sites i ¼ 1; 2 . . . N 1 are occupied by T ions, and site i ¼ N is filled by the T 0 one. Furthermore, the z-axis is chosen along the perpendicular to the ring plane, and the local second-order ZFS is dominated by the axial terms di , with smaller rhombic terms ei . Fig. 2 shows the INS intensity for a Cr7Ni sample at T ¼ 2 K, recorded in zero magnetic field with k ¼ 5 Å. The two peaks (at 1.19 meV and 1.34 meV, respectively) can be attributed to transitions between the j S ¼ 1=2; M ¼ 1=2i ground state and the anisotropy split sublevels of the j S ¼ 3=2i first excited spin multiplet, j S ¼ 3=2; M ¼ 1=2i and j S ¼ 3=2; M ¼ 3=2i. The experimental data are well reproduced by the calculations (Eq. 9), with the parameters J Cr—Cr ¼ 1:46 meV, J Cr—Ni ¼ 1:69 meV, dNi ¼ 0:35 meV, dCr ¼ 0:03 meV, and negligible ei values. The value of dCr has been kept fixed to that determined for the homonuclear Cr8 ring [35], which shares the same local structure as Cr7Ni. As a result of these calculations, the admixture of j S ¼ 3=2i components in the j S ¼ 1=2i ground state is extremely small (about 1%); nevertheless, S-mixing must be taken into account to reproduce correctly the intensity ratio of the observed doublet. Transitions from the ground state to j S ¼ 1=2i and j S ¼ 3=2i excited states have also been observed at higher energy transfers, and are in excellent agreement with the calculations. When an external magnetic field B with appropriate value and direction is applied, several level crossings turn into ‘‘avoided crossings” (ACs), as a result of the S-mixing. The AC gap D is maximum when the direction of the applied field forms an angle
G. Amoretti et al. / Inorganica Chimica Acta 361 (2008) 3771–3776
Fig. 2. Low energy transfer INS response for Cr7Ni at 2 K ðk ¼ 5 ÅÞ. Background from the sample holder has been subtracted. Solid line: intensity calculated for the model Hamiltonian (9). A schematic representation of the Cr7Ni molecule is shown as inset; Cr (green spheres), Ni (blue sphere), F (yellow spheres), O (red spheres), C (black dots); H atoms not shown. From Ref. [9].
h ¼ p=4 with the z-axis, whereas it vanishes for h ¼ 0 and h ¼ p=2. At the AC occurring near 10.5 T, the ground-state wavefunction is given by the superposition of two different spin states j S; MS > (with the quantization axis along the field direction): 3 1 1 1 3 j wi ¼ pffiffiffi ; ; ; ð10Þ 2 2 2 2 2 and the total spin of the ring oscillates in time between 1/2 and 3/2 with the frequency D= h [36,37]. The occurrence of these coherent quantum oscillations has been directly observed by one of the few INS experiments on a single crystal of magnetic molecules up to now. Measurements were performed at 66 mK, so that only the ground state was populated even at the AC. The experiment shows indeed a resonant inelastic peak in the cross-section at an energy corresponding to D, with a field-dependent width that reaches a minimum (the experimental resolution) at the AC. These results demonstrate that the total spin of the molecule oscillates coherently for a number of cycles much larger than unity, as determined by the ratio between the oscillation frequency and the intrinsic width [31]. Another precise signature of the AC is present in the INS spectrum in the 0.6–1.6 meV energy range. When kT is much smaller than D, spectra taken at the AC field BAC are expected to contain in this energy range sizeable extra contributions, which reflect transitions from the ground state to excited S-multiplets. These transitions are switched on at the AC due to the mixed nature of the ground state. By collecting spectra for B slightly less than BAC , and for B ¼ BAC , these extra contributions can be readily identified, providing new important quantitative information on the groundstate wavefunction. The agreement between theory and experiment is extremely good, as shown in Fig. 3.
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Fig. 3. Intensity plot showing the energy and magnetic field dependence of the calculated T = 66 mK INS cross-section of Cr7Ni, for an angle h ¼ 50 between the magnetic field and the normal to the ring plane. Points indicate the positions of the experimentally observed INS peaks. From Ref. [31].
hnz ðtÞnz ð0Þi ¼
X
eiDm t=h j hm j nz j 0ij2 ;
ð11Þ
m
should oscillate harmonically: all the spectral weights j hm j nz j 0ij2 should be zero except j h1 j nz j 0ij2 which should be equal to s2 (25/ 4 in the case of Fe3+). The NVT is the AF counterpart of the tunneling of the molecular magnetization M characterizing nanomagnets such as Mn12 or Fe8. For the latter the tunneling time is macroscopic, and so long that the time autocorrelation of M becomes overdamped due to dissipation into the nuclear-spins subsystem. Thus, the temporal oscillations of M associated with coherent tunneling do not actually take place. On the contrary, the time-scale of NVT in AF rings is microscopic, and much shorter than the decoherence time associated with spin–nuclei or spin–phonon interactions. This time-scale is just in the typical range accessed by cold neutrons. Moreover, INS is the technique of choice to study NVT since the neutron cross-section directly probes the autocorrelation function of n. This allows one to determine to which extent the autocorrelation is monochromatic (as in the NVT picture discussed above), and to measure the value of the tunnelling frequency.
4.2. Tunneling of the Néel vector An interesting aspect of the dynamics of AF rings is the possible occurrence of tunneling of the direction of the Néel vector P n ¼ Ni¼1 ð1Þi si =N (NVT) [11,38,39]. In the NVT regime, n would tunnel coherently between the two degenerate ±z directions. The two lowest energy states of the system would be even pffiffiffi and odd combinations of the classical Néel states (i.e. 1= 2ðj"; #; "; . . .i j#; "; #; . . .i)), and the T ¼ 0 time correlation functions, which has the general form
Fig. 4. Inelastic neutron scattering spectra for Fe7Mn collected at different temperatures. Solid lines are spectra calculated as described in the text. From Ref. [40].
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References
Fig. 5. Magnetic field dependence of the spectral weights of the zero-temperature autocorrelation function describing the Néel vector dynamics in Fe7Mn. The black line is the weight of the fundamental component, j h1 j nz j 0ij2 , the red one is the sum of the higher harmonics spectral weights. Calculations are based on the Hamiltonian (9), with parameters deduced from INS measurements. A Zeeman term corresponding to a magnetic field in the ring plane has been added to Eq. (9). Broken lines represent the results of calculations with S-mixing forced to zero. From Ref. [40].
A recent investigation of this phenomenon has been performed on the Fe7Mn ring, containing Fe3+ ions ðs ¼ 5=2Þ and one Mn2+ ion in the high-spin s ¼ 5=2 state [40]. The total spin of the ring in the ground state is therefore S ¼ 0. INS spectra collected at different temperatures on a polycrystalline sample are shown in Fig. 4. The data have been interpreted using Hamiltonian (9), with the following site-independent parameters: J n:n: ¼ 1:42 meV (nearestneighbor exchange), d ¼ 15 leV, j e=d j¼ 0:1 (ZFS), and a small next-nearest neighbor isotropic exchange contribution J n:n:n: ¼ 20 leV. Using these parameters, the magnetic-field dependence of the autocorrelation function has been calculated. The results are shown in Fig. 5, where the spectral weight j h1 j nz j 0ij2 is compared with the sum of all other components. Although the oscillations are almost monochromatic for B ¼ 0 and for values of B intermediate between two ground-state crossing fields (where the black line has maxima), the NVT regime is not achieved since j h1 j nz j 0ij2 is less than half the expected s2 value. It is worth to note that the fluctuation of the total spin plays an important role also in the Néel vector dynamics. In fact, if S mixing is artificially suppressed, the autocorrelation of nz is not characterized by a single frequency for B 6¼ 0 (see Fig. 5). 5. Conclusions The results discussed in this short review represent but a small fraction of the large number of INS studies of molecular nanomagnets. This extensive amount of work has been stimulated by the important discoveries of the group of Dante Gatteschi in Florence, which proved to be exceedingly fertile in fundamental research and promise to open new frontiers in technology. Following our first study on the Fe8 cluster in collaboration with Dante’s group, we have enjoyed a fruitful and pleasant collaboration over these years. We are deeply indebted to him for the excellent opportunities he has offered us.
[1] D. Gatteschi, R. Sessoli, Angew. Chem., Int. Ed. Engl. 42 (2003) 268. and references therein. [2] M.N. Leuenberger, D. Loss, Nature 410 (2001) 789. [3] S. Carretta, P. Santini, G. Amoretti, F. Troiani, M. Affronte, Phys. Rev. B 76 (2007) 024408. [4] F. Troiani, M. Affronte, S. Carretta, P. Santini, G. Amoretti, Phys. Rev. Lett. 94 (2005) 190501. [5] F. Troiani, A. Ghirri, M. Affronte, S. Carretta, P. Santini, G. Amoretti, S. Piligkos, G. Timco, R.E.P. Winpenny, Phys. Rev. Lett. 94 (2005) 207208. [6] E. Liviotti, S. Carretta, G. Amoretti, J. Chem. Phys. 117 (2002) 3361. [7] R. Caciuffo, G. Amoretti, A. Murani, R. Sessoli, A. Caneschi, D. Gatteschi, Phys. Rev. Lett. 81 (1998) 4744. [8] I. Mirebeau, M. Hennion, H. Casalta, H. Andres, H.U. Gudel, A.V. Irodova, A. Caneschi, Phys. Rev. Lett. 83 (1999) 628. [9] R. Caciuffo, T. Guidi, G. Amoretti, S. Carretta, E. Liviotti, P. Santini, C. Mondelli, G. Timco, C.A. Muryn, R.E.P. Winpenny, Phys. Rev. B 71 (2005) 174407. [10] J.T. Haraldsen, T. Barnes, J.L. Musfeldt, Phys. Rev. B 71 (2005) 064403. [11] P. Santini, S. Carretta, G. Amoretti, T. Guidi, R. Caciuffo, A. Caneschi, D. Rovai, Y. Qiu, J.R.D. Copley, Phys. Rev. B 71 (2005) 184405. [12] O. Waldmann, C. Dobe, H.U. Gudel, H. Mutka, Phys. Rev. B 74 (2006) 054429. [13] S. Carretta, P. Santini, G. Amoretti, T. Guidi, J. Dyson, R. Caciuffo, J.A. Stride, A. Caneschi, J.R.D. Copley, Phys. Rev. B 73 (2006) 144425. [14] S.T. Ochsenbein, O. Waldmann, A. Sieber, G. Carver, R. Bircher, H.U. Güdel, R.S.G. Davies, G.A. Timco, R.E.P. Winpenny, H. Mutka, et al., EPL (Europhys. Lett.) 79 (2007) 17003. [15] M.B. Stone, F. Fernandez-Alonso, D.T. Adroja, N.S. Dalal, D. Villagran, F.A. Cotton, S.E. Nagler, Phys. Rev. B 75 (2007) 214427. [16] O. Waldmann, R. Bircher, G. Carver, A. Sieber, H.U. Gudel, H. Mutka, Phys. Rev. B 75 (2007). [17] O. Waldmann, G. Carver, C. Dobe, A. Sieber, H. Gudel, H. Mutka, J. Am. Chem. Soc. 129 (2007) 1526. [18] G. Amoretti, S. Carretta, R. Caciuffo, H. Casalta, A. Cornia, M. Affronte, D. Gatteschi, Phys. Rev. B 64 (2001) 104403. [19] T. Guidi, S. Carretta, P. Santini, E. Liviotti, N. Magnani, C. Mondelli, O. Waldmann, L.K. Thompson, L. Zhao, C.D. Frost, et al., Phys. Rev. B 69 (2004) 104432. [20] S.W. Lovesey, Theory of Neutron Scatteirng from Condensed Matter, vols. 1 and 2, Clarendon Press, Oxford, 1987. [21] J.J. Borras-Almenar, J.M. Clemente-Juan, E. Coronado, B.S. Tsukerblat, Inorg. Chem. 38 (1999) 6081. [22] O. Waldmann, Phys. Rev. B 68 (2002) 174406. [23] S. Carretta, E. Liviotti, N. Magnani, P. Santini, G. Amoretti, Polyhedron 24 (2005) 2459. [24] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature 383 (1996) 145. [25] W. Wernsdorfer, R. Sessoli, Science 284 (1999) 133. [26] G. Amoretti, R. Caciuffo, J. Combet, A. Murani, A. Caneschi, Phys. Rev. B 62 (2000) 3022. [27] S. Carretta, E. Liviotti, N. Magnani, P. Santini, G. Amoretti, Phys. Rev. Lett. 92 (2004) 207205. [28] A. Cornia, A.C. Fabretti, P. Garrisi, C. Mortalò, D. Bonacchi, D. Gatteschi, R. Sessoli, L. Sorace, W. Wernsdorfer, A.L. Barra, Angew. Chem., Int. Ed. 43 (2004) 1136. [29] S. Carretta, P. Santini, G. Amoretti, T. Guidi, R. Caciuffo, A. Candini, A. Cornia, D. Gatteschi, M. Plazenet, J.A. Stride, Phys. Rev. B 70 (2004) 214403. [30] M. Affronte, A. Ghirri, S. Carretta, G. Amoretti, S. Piligkos, G.A. Timco, R.E.P. Winpenny, Appl. Phys. Lett. 84 (2004) 3468. [31] S. Carretta, P. Santini, G. Amoretti, T. Guidi, J.R.D. Copley, Y. Qiu, R. Caciuffo, G. Timco, R.E.P. Winpenny, Phys. Rev. Lett. 98 (2007) 167401. [32] J. Schnack, M. Luban, Phys. Rev. B 63 (2001) 014418. [33] O. Waldmann, T. Guidi, S. Carretta, C. Mondelli, A.L. Dearden, Phys. Rev. Lett. 91 (2003) 237202. [34] F.K. Larsen, J. Overgaard, S. Parsons, E. Rentschler, A.A. Smith, G.A. Timco, R.E.P. Winpenny, Angew. Chem., Int. Ed. 42 (2003) 5978. [35] S. Carretta, J. van Slageren, T. Guidi, E. Liviotti, C. Mondelli, D. Rovai, A. Cornia, A.L. Dearden, M. Affronte, C.D. Frost, et al., Phys. Rev. B 67 (2003) 094405. [36] S. Carretta, P. Santini, E. Liviotti, N. Magnani, T. Guidi, R. Caciuffo, G. Amoretti, Eur. Phys. J. B 36 (2003) 169. [37] O. Waldmann, S. Carretta, P. Santini, R. Koch, A.G.M. Jansen, G. Amoretti, R. Caciuffo, L. Zhao, L.K. Thompson, Phys. Rev. Lett. 92 (2004) 096403. [38] F. Meier, D. Loss, Phys. Rev. B 64 (2001) 224411. [39] A. Chiolero, D. Loss, Phys. Rev. Lett. 80 (1998) 169. [40] T. Guidi, J.R.D. Copley, Y. Qiu, S. Carretta, P. Santini, G. Amoretti, G. Timco, R.E.P. Winpenny, C.L. Dennis, R. Caciuffo, Phys. Rev. B 75 (2007) 014408.
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Inorganica Chimica Acta journal homepage: www.elsevier.com/locate/ica
Review
Inelastic neutron scattering investigations of molecular nanomagnets G. Amoretti a, R. Caciuffo b,*, S. Carretta a, T. Guidi c, N. Magnani b, P. Santini a a b c
Dipartimento di Fisica, Università di Parma, I-43100 Parma, Italy European Commission, Joint Research Centre, Institute for Transuranium Elements, Postfach 2340, D-76125 Karlsruhe, Germany Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin, Germany
a r t i c l e
i n f o
Article history: Received 29 January 2008 Received in revised form 4 March 2008 Accepted 6 March 2008 Available online 16 March 2008 Dedicated to Dante Gatteschi. PACS: 75.50.Tt 75.10.Jm 75.40.Gb 75.45.+j
a b s t r a c t Inelastic neutron scattering, probing the temporal spin–spin correlation at the microscopic scale, is a powerful technique to study the magnetic behaviour of molecular nanomagnets. Experiments at different energy scales and different energy-transfer resolution allow precise determinations of the parameters defining the effective Hamiltonians used to model the diverse physical properties exhibited by this class of materials. The intrinsic disadvantage of the technique (low flux, requiring a sample mass in the gram scale) is over-compensated by the large amount of information that can be straightforwardly extracted. Zero-field splittings and exchange interactions can be determined with a large degree of confidence, shedding light on important issues such as magnetization tunnelling in giant-spin clusters, or the occurrence of quantum coherence phenomena and their consequences on macroscopic observables. Ó 2008 Elsevier B.V. All rights reserved.
Keywords: Inelastic neutron scattering Molecular nanomagnets Spin hamiltonian
Giuseppe Amoretti is Full Professor of experimental physics and Head of the Physics Department of the University of Parma. In 1982/83 he worked two years at the Département de Recherche Fondamentale, Centre d’Etudes Nucléaires de Grenoble, with a fellowship of the European Institute for Transuranium Elements in Karlsruhe, Germany. He was the coordinator of the INFM (D Section) network on ‘‘Fundamental Magnetic Interactions”. He was member of the ‘‘Board of Directors of the European Actinide Research Development Society”. He is the author or coauthor of about 140 publications concerning the physics of condensed matter, in particular the properties of rare-earth, actinide and transition metal compounds. His current activity is mainly focused on molecular magnetism. His contribution is primarily theoretical and is related to the design and the interpretation of experiments performed by neutrons or by other spectroscopic and bulk techniques (EPR, NMR, NQR, lSR, Mössbauer, magnetic measurements, specific heat). Since 1986 he has performed many neutron scattering experiments, at RAL (UK) and ILL (France).
* Corresponding author. Tel.: +49 7247951382; fax: +49 7247951599. E-mail address: [email protected] (R. Caciuffo). 0020-1693/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ica.2008.03.047
Roberto Caciuffo is Head of the Actinide Research Department at the European Commission’s Institute for Transuranium Elements in Karlsruhe, Germany, and Full Professor of experimental physics at the Università Politecnica delle Marche, Ancona, Italy. He has previously been an experimental physicist at the Institut LaueLangevin (France), at the Centre d’Etudes Nucléaires de Grenoble (France), and at the Rutherford Appleton Laboratory (UK). He is a member of advisory boards of several international institutions. His current research is focussed on the study of magnetic correlations and multipolar order in d- and f-electron systems by neutron and synchrotron radiation X-ray scattering. He is author or coauthor of over 200 publications in leading international journals.
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Stefano Carretta obtained both his degrees from University of Parma (Italy), doing his PhD research on molecular magnetism with Prof. G. Amoretti. In 2005 he has been research fellow of the National Institute for the Physics of Matter (INFM) and at the Faculty of Science at Parma. His research has been mainly focused on theoretical modelling of the quantum behaviour of magnetic molecules. The aim of this research is twofold: the understanding of fundamental phenomena like quantum tunnelling and the identification of systems suitable for technological applications, especially in the field of Quantum Computation. He works in close collaborations with experimental groups, and he actively participates to neutron scattering experiments. He has published more than 50 research papers. In 2006 he has been awarded the ”Le Scienze” and President of the Italian Republic medal for his research on molecular nanomagnets.
Tatiana Guidi graduated in physics in 2001 from the University of Camerino, Italy. She obtained her PhD in Material Science Engineering at the Università Politecnica delle Marche, Italy, studying spin excitations and magnetic properties of Molecular Nanomagnets, using neutron scattering technique. She continued her activity there as a post Doc and spent one year as a guest researcher at NIST Center for Neutron Research (USA). She then joined the Junior Research Group ‘‘Magnetism and Superconductivity of Quantum Materials” of Hahn-Meitner-Institut, Berlin as a Post Doc (2006–2008). Since January 2008 she has been appointed as instrument scientist in the excitation group of the ISIS spallation source, Rutherford Appleton Laboratory, UK, where she will continue with her activity in the field of molecular nanomagnetism. She has published more than 20 papers in international peer-reviewed journals.
Nicola Magnani was born in 1976. He graduated in Physics in the year 2000 at the University of Parma (Italy), where he also obtained his PhD in 2003. After post-doc experiences in the field of molecular nanomagnetism and magnetocaloric effect, since 2006 he holds a Research Fellow position at the at the European Commission’s Institute for Transuranium Elements in Karlsruhe (Germany). His research interests mainly concern the fundamental magnetic properties of d- and f-electron systems, both from the theoretical point of view (crystal-field theory) and by several experimental techniques (mainly inelastic neutron scattering). He published more than 50 papers in international peer-reviewed journals.
Paolo Santini got his PhD in Physics at the University of Lausanne, Switzerland (1995), where he continued until 1999 as lecturer and Assistant Professor. From 1999 to 2002 he worked at the Clarendon Laboratory (Oxford), with funds provided by the Swiss National Science Foundation. In 2002 he returned to Italy with a grant funded by the Italian government to reduce the Italian ‘‘brain-drain”. Since 2006 he has been Associate Professor of condensed matter physics at the University of Parma. His research is in theoretical condensed matter physics: strongly correlated electron systems, magnetism in rare-earth and actinide compounds, low-dimensional systems such as spinchains or semiconductor surfaces, and more recently molecular magnets. He has been involved both in fundamental problems and in the investigation of specific materials. For the latter aspect he has established collaborations with several experimental groups, and he regularly visits large-scale facilities to participate in neutron scattering experiments. He is author or coauthor of about 70 publications in leading international journals.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inelastic neutron scattering cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero-field splitting and tunnelling of the magnetization in single molecule magnets . . . . . Antiferromagnetic molecular rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Quantum oscillations of the total spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Tunneling of the Néel vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.
1. Introduction Following the discovery of slow magnetic relaxation and quantum tunneling in Mn12, molecular nanomagnets (MNM) have attracted considerable attention from physicists and chemists alike [1]. Unique applications have been envisaged as ultra-high density memory devices, or for the implementation of quantum computing algorithms exploiting the superposition of single particle quantum states [2–5]. In the absence of reliable ab-initio calculation methods, the physics of MNM is conveniently described by an effective spin Hamiltonian containing all the relevant intra-cluster interactions, that is the Heisenberg exchange, the single-ion zero-field splitting (ZFS), the magnetic dipole–dipole interaction, and the Zeeman term (in presence of an external magnetic field B): H¼
X
J ij si sj þ
i>j¼1;N
þ
X i>j¼1;N
X X i¼1;N
q ^ q ðsi Þ bk ðiÞO k
si Dij sj lB
k
i¼1;N
~ðiÞ si : Bg
ð1Þ
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3772 3773 3773 3774 3774 3775 3776 3776
Here, N is the number of magnetic centres forming the cluster, J ij are ^ q ðsi Þ are Stevens operator equivalents of pair exchange integrals, O k q rank k (k even and 64 for d electrons), bk ðiÞ are ZFS coefficients, ~ ðiÞ are g-factor tenDij is the dipole–dipole interaction tensor, and g sors. The spin Hamiltonian, which phenomenologically describes the magnetic interactions within each molecule, is widely used in molecular magnetism. This approach eliminates all the orbital coordinates of the system and allows to easily take advantage of the molecular symmetry. In particular, the number of non-zero indeq pendent bk ðiÞ parameters is restricted by group theory. Where the Heisenberg term is dominant, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in the so-called strong-exchange limit, the modulus SðS þ 1Þ of the total spin P S ¼ i si is practically conserved and the energy spectrum is formed by multiplets of defined S. In this approximation, the anistropic part of (1) can be projected on a given multiplet (usually the ground state), leading to XX q q ^ ðSÞ l B g ~ S; HZFS ¼ Bk O ð2Þ B k
k;q
X
... ... ... ... ... ... ... ...
Bqk
q
with parameters calculated by projection coefficients from the q bk ðiÞ and Dij parameters appearing in Eq. (1).
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The anisotropic part of the spin Hamiltonian (1) can be written in terms of irreducible tensor operators (ITO) of rank k P 1. Thus a mixing of states with different S and M, or at least states with different S if the anisotropy is purely axial, is expected. Difficulties arising from S-mixing can be dealt with the procedure described in [6]. The parameters defining the Hamiltonians (1) and (2) must be considered as phenomenological quantities to be determined from experiment. For this purpose, inelastic neutron scattering (INS) is a very useful technique, in that it gives access to eigenvalues and eigenstates through the energy and the intensity of the resonances appearing in the scattering cross-section. Several applications of INS to the study of MNM have been reported in the literature, often giving information crucial to understand the macroscopic behaviour of the investigated systems [7–19]. In the following, after a brief presentation of the appropriate INS cross-section formulae, a few selected examples showing the potential of the technique will be reviewed. 2. Inelastic neutron scattering cross-section The double-differential magnetic cross-section for unpolarised neutrons is given by [20] X X o2 r kf ^ aQ ^ bÞ ¼ ðcN r 0 Þ2 e2W ðda;b Q g j F j ðQ Þg ‘ F ‘ ðQ Þ expðiQ Rj‘ Þ oXoEf ki a;b j;‘ X pm hm j saj j lihl j sb‘ j midðhx þ Em El Þ; ð3Þ m;l
where a; b are Cartesian coordinates, saj is the a component of the spin operator of the jth ion, j mi is the eigenfunction with energy Em and occupation probability pm . We use the conventional definition of scattering vector Q ¼ ki kf and energy transfer hx ¼ Ei Ef , where EiðfÞ and kiðfÞ are the incident (final) neutron energy and wavevector, respectively; cN is the neutron magnetic dipole moment in nuclear Bohr magnetons, r 0 is the classical electron radius, g j and F j ðQ Þ are the Landé factor and the dipolar form factor of the jth ion, and e2W is the Debye–Waller factor. The intrinsic linewidth of the excitations and the finite instrument resolution are taken into account by replacing the d function in Eq. (3) by an appropriate spectral weight function, usually a Gaussian or a Lorentzian line-shape. For polycrystalline or powder samples, Eq. (3) must be averaged with respect to the possible orientations of the scattering vector Q. An expression of the resulting average adequate for systems with small magnetic anisotropy is given in [21], whereas one valid for uniaxial anisotropy (regardless of its magnitude) and properly accounting for intramolecular interference effects is reported in [22]. A formula of general validity is [9]: o2 r A kf 2W X ¼ e pm Iml ðQ Þdð hx El þ Em Þ; oXoEf N m ki m;l
ð4Þ
where A ¼ 0:29 barn, N m is the number of magnetic ions, and the function Iml (Q) is defined as h i X 2 j0 ðQRj‘ Þ þ C 20 j2 ðQRj‘ Þ ~szj ~sz‘ Iml ðQ Þ ¼ F j ðQ ÞF ‘ ðQ Þ 3 j;‘ 2 1 þ j0 ðQRj‘ Þ C 20 j2 ðQRj‘ Þ ð~sxj ~sx‘ þ ~syj ~sy‘ Þ 3 2 1 2 ~ þ j2 ðQRj‘ Þ½C 2 ðsxj ~sx‘ ~syj ~sy‘ Þ þ C 22 ð~sxj ~sy‘ þ ~syj ~sx‘ Þ 2 o þ j2 ðQRj‘ Þ½C 21 ð~szj ~sx‘ þ ~sxj ~sz‘ Þ þ C 21 ð~szj ~sy‘ þ ~syj ~sz‘ Þ ; ð5Þ where Rj‘ gives the relative position of ions j and ‘,
C 20 ¼ C 22 ¼
2 1 Rj‘z ½3 1; 2 Rj‘ R2j‘x R2j‘y R2j‘
C 22 ¼ 2 C 21 ¼
Rj‘x Rj‘y R2j‘
Rj‘x Rj‘z
C 21 ¼
R2j‘
;
;
Rj‘y Rj‘z R2j‘
;
ð6Þ
;
and ~saj ~sc‘ ¼ hm j saj j lihl j sc‘ j mi
ða; c ¼ x; y; zÞ:
ð7Þ
The INS spectra are therefore composed by a discrete number of peaks, whose intensity is a linear combination of products of geometrical factors (depending on the momentum transfer Q) and matrix elements (between initial and final states) of the cartesian components of the spin operators sj . The Hamiltonian (1) can be used to calculate Em and all of the ~saj . A comparison between the experimental spectra and the calculated counterparts allows then determining the parameters of the microscopic Hamiltonian. A stringent test for the spectroscopic assignment of the observed transitions is provided by the Q dependence of their intensity, which is essentially determined by the geometry of the cluster and the composition of the spin wavefunctions. This dependence can be easily measured with properly calibrated detectors at different scattering angles. Where the strong exchange limit holds, Eq. (4) can be replaced by a much simpler expression involving the matrix elements of the cartesian components of the total spin S [23]: X X o2 r / pm j hm j Sa j lij2 dð hx El þ Em Þ: oXoEf m;l a¼x;y;z
ð8Þ
This is the same cross-section widely used for the interpretation of crystal-field excitations measured for polycrystalline samples, with the total spin operator replacing the total angular momentum. Its validity in the present case is limited to molecules with uniaxial anisotropy (z being the symmetry direction), or when Q ’ 0.
3. Zero-field splitting and tunnelling of the magnetization in single molecule magnets Quantum tunnelling of the magnetization in molecules with a magnetic ground state and a high Ising-like anisotropy barrier has been widely investigated in the last decade [24,25]. The phenomenon is allowed in presence of ZFS terms breaking the axial symmetry. In that case, the total spin component along the anisotropy axis z is time dependent and the spin state oscillates with time between j S; Si and j S; Si. The degeneracy of the two states associated to opposite sides of the anisotropy barrier is removed by tunnelling, and the tunnel splitting D determines the time dependence of the low temperature magnetic relaxation. The value of D can be estimated quite precisely from magnetization measurements, but it is usually too small to be accessible by INS. For instance, D is of the order of 108 meV in the case of the Fe8Br cluster. Nevertheless, INS leads to a determination of the ZFS Hamiltonian, from which the tunnel splitting can be easily calculated by numerical diagonalization. This is true, provided that all the ingredients of the physical problem are correctly taken into account. An instructive example is that of Fe8Br, one of the first giant-spin MNM studied by neutrons [7,26]. The molecule is characterized by an approximate D2 symmetry. Antiferromagnetic interactions
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relaxation time sðBÞ occur in correspondence of level anti-crossing at B ¼ 0:5 and 1 T. This behaviour is correctly reproduced by the ZFS Hamiltonian derived from the INS results, the trigonal term ^ 3 being responsible for anti-crossing at 0.5 T, and O ^ 2 for the O 4 2 anti-crossing at 1 T.
20
15
10
Energy (meV)
Intensity (arb. units)
3
2
4. Antiferromagnetic molecular rings
1
4.1. Quantum oscillations of the total spin 0
5
0 -0.4
-0.2
0
0.2
0.4
Energy Transfer (meV) Fig. 1. INS spectrum of Fe8Br measured at 9.6 K. The well defined peaks correspond to dipole-allowed transitions between the zero-field split components of the S ¼ 10 ground state, schematised in the inset. The smooth line is the theoretical spectrum for the ZFS Hamiltonian in the strong-exchange limit. From Ref. [7].
between Fe ions lead to a S ¼ 10 ground state, approximately described by eight individual spins s ¼ 5=2, six parallel to each other and two antiparallel. The high-resolution INS spectrum of Fe8Br, taken at T ¼ 9:6 K with an energy resolution of 19 leV at the elastic peak, is shown in Fig. 1. The peaks in the cross-section correspond to transitions between components of the S = 10 ground state, split by a ZFS interaction of D2 symmetry. The analysis of the experimental spectrum with Eqs. 2 and 8 gives a precise estimate of the allowed parameters, B02 ¼ 8:4ð1Þ leV, B22 ¼ 4:02ð3Þ leV, 4 4 0 2 B4 ¼ 0:87ð6Þ10 leV, B4 ¼ 0:1ð1Þ10 leV and B44 ¼ 7:4ð6Þ 104 leV. However, when the Hamiltonian (2) is diagonalised with these set of parameters, one obtains a tunnel splitting of 3:8 1012 meV, four orders of magnitude smaller than the value given by macroscopic measurements. This huge discrepancy reflects the inadequacy of the strong-exchange limit. Indeed, the first excited spin state, a S ¼ 9 multiplet, is about 4 meV above the ground state, and some degree of S-mixing is expected. Although small, S-mixing is very effective in enhancing the tunnel splitting, as it opens highly efficient tunnelling channels. When this fact is properly taken into account, a value of D consistent with that deduced from magnetic relaxation measurements can be obtained [27]. In many cases, INS experiments can lead to a determination of the point symmetry of the cluster in the low-temperature phase, with a sensitivity not parallelled by diffraction measurements. An example is given by the study of the Fe4 cluster [28], a molecule formed by three Fe3+ ions lying at the vertices of an equilateral triangle and by one Fe3+ ion at its centre. X-ray diffraction at room temperature suggests a D3 symmetry, with the C 3 axis perpendicular to the plane of the equilateral triangle. The ground state has a total spin S ¼ 5, with the two lowest quasi-degenerate states approximately given by j S ¼ 5; M ¼ 5i. Intramultiplet excitations have been measured by high-resolution INS experiments ^ 2 term, [29]. A good fit of the results is obtained only if a rhombic O 2 not allowed by the D3 symmetry, is introduced. This term is necessary to account for the manifestation of zero-field quantum tunnelling effects in the magnetic relaxation below 0.2 K [28]. In fact, the tunnel splitting D would be zero if the only non-axial ZFS term ^ 3 , since j S ¼ 5; M ¼ 5i are not were the D3 symmetry-allowed O 4 ^ 3 . Conversely, when a magnetic field connected by any power of O 4 is applied, shortcuts for the tunneling process and minima in the
Ring-shaped cyclic clusters formed by coplanar antiferromagnetically coupled ions have attracted considerable attention for the peculiar low-energy spin dynamics they exhibit and the potential applications as qubits or low-temperature magnetocaloric refrigerants [30]. The possibility to observe quantum tunnelling of the Néel vector in these systems has been extensively debated, and the occurrence of quantum oscillations of the total spin length at anti-crossings between different S multiplets has been clearly demonstrated [31]. For an even number of spins, the ground state has S ¼ 0 and the excitation spectrum is characterized by rotational and spin-wave like bands [32,33]. This is the case of the homometallic Cr8 ring, formed by octahedrally coordinated Cr3+ ions with s ¼ 3=2. Heterometallic rings with S 6¼ 0 can be obtained from a S ¼ 0 homonuclear ring by chemical substitution of one or two magnetic centres [34,35]. Such a substitution affects also the spin topology and the overall macroscopic behaviour of the system [9]. For instance, substitution of Cr3+ with Ni2+, gives a cluster with spin ground state S ¼ 1=2, which could be a suitable candidate for the physical implementation of qubits [5]. For a ring-shaped molecule containing N 1 magnetic centers T and one magnetic centre T 0 , the spin Hamiltonian (1) assumes the form H¼
N 2 X
J TT si siþ1 þ J TT 0 ðsN1 sN þ sN s1 Þ þ
i¼1
þ
N X i¼1
N X
di ½s2z;i
1 si ðsi þ 1Þ 3
gðiÞB si :
ð9Þ
i¼1
ei ½s2x;i
s2y;i
þ
X i>j¼1;N
si Dij sj lB
X i¼1;N
In the above equation it is assumed that sites i ¼ 1; 2 . . . N 1 are occupied by T ions, and site i ¼ N is filled by the T 0 one. Furthermore, the z-axis is chosen along the perpendicular to the ring plane, and the local second-order ZFS is dominated by the axial terms di , with smaller rhombic terms ei . Fig. 2 shows the INS intensity for a Cr7Ni sample at T ¼ 2 K, recorded in zero magnetic field with k ¼ 5 Å. The two peaks (at 1.19 meV and 1.34 meV, respectively) can be attributed to transitions between the j S ¼ 1=2; M ¼ 1=2i ground state and the anisotropy split sublevels of the j S ¼ 3=2i first excited spin multiplet, j S ¼ 3=2; M ¼ 1=2i and j S ¼ 3=2; M ¼ 3=2i. The experimental data are well reproduced by the calculations (Eq. 9), with the parameters J Cr—Cr ¼ 1:46 meV, J Cr—Ni ¼ 1:69 meV, dNi ¼ 0:35 meV, dCr ¼ 0:03 meV, and negligible ei values. The value of dCr has been kept fixed to that determined for the homonuclear Cr8 ring [35], which shares the same local structure as Cr7Ni. As a result of these calculations, the admixture of j S ¼ 3=2i components in the j S ¼ 1=2i ground state is extremely small (about 1%); nevertheless, S-mixing must be taken into account to reproduce correctly the intensity ratio of the observed doublet. Transitions from the ground state to j S ¼ 1=2i and j S ¼ 3=2i excited states have also been observed at higher energy transfers, and are in excellent agreement with the calculations. When an external magnetic field B with appropriate value and direction is applied, several level crossings turn into ‘‘avoided crossings” (ACs), as a result of the S-mixing. The AC gap D is maximum when the direction of the applied field forms an angle
G. Amoretti et al. / Inorganica Chimica Acta 361 (2008) 3771–3776
Fig. 2. Low energy transfer INS response for Cr7Ni at 2 K ðk ¼ 5 ÅÞ. Background from the sample holder has been subtracted. Solid line: intensity calculated for the model Hamiltonian (9). A schematic representation of the Cr7Ni molecule is shown as inset; Cr (green spheres), Ni (blue sphere), F (yellow spheres), O (red spheres), C (black dots); H atoms not shown. From Ref. [9].
h ¼ p=4 with the z-axis, whereas it vanishes for h ¼ 0 and h ¼ p=2. At the AC occurring near 10.5 T, the ground-state wavefunction is given by the superposition of two different spin states j S; MS > (with the quantization axis along the field direction): 3 1 1 1 3 j wi ¼ pffiffiffi ; ; ; ð10Þ 2 2 2 2 2 and the total spin of the ring oscillates in time between 1/2 and 3/2 with the frequency D= h [36,37]. The occurrence of these coherent quantum oscillations has been directly observed by one of the few INS experiments on a single crystal of magnetic molecules up to now. Measurements were performed at 66 mK, so that only the ground state was populated even at the AC. The experiment shows indeed a resonant inelastic peak in the cross-section at an energy corresponding to D, with a field-dependent width that reaches a minimum (the experimental resolution) at the AC. These results demonstrate that the total spin of the molecule oscillates coherently for a number of cycles much larger than unity, as determined by the ratio between the oscillation frequency and the intrinsic width [31]. Another precise signature of the AC is present in the INS spectrum in the 0.6–1.6 meV energy range. When kT is much smaller than D, spectra taken at the AC field BAC are expected to contain in this energy range sizeable extra contributions, which reflect transitions from the ground state to excited S-multiplets. These transitions are switched on at the AC due to the mixed nature of the ground state. By collecting spectra for B slightly less than BAC , and for B ¼ BAC , these extra contributions can be readily identified, providing new important quantitative information on the groundstate wavefunction. The agreement between theory and experiment is extremely good, as shown in Fig. 3.
3775
Fig. 3. Intensity plot showing the energy and magnetic field dependence of the calculated T = 66 mK INS cross-section of Cr7Ni, for an angle h ¼ 50 between the magnetic field and the normal to the ring plane. Points indicate the positions of the experimentally observed INS peaks. From Ref. [31].
hnz ðtÞnz ð0Þi ¼
X
eiDm t=h j hm j nz j 0ij2 ;
ð11Þ
m
should oscillate harmonically: all the spectral weights j hm j nz j 0ij2 should be zero except j h1 j nz j 0ij2 which should be equal to s2 (25/ 4 in the case of Fe3+). The NVT is the AF counterpart of the tunneling of the molecular magnetization M characterizing nanomagnets such as Mn12 or Fe8. For the latter the tunneling time is macroscopic, and so long that the time autocorrelation of M becomes overdamped due to dissipation into the nuclear-spins subsystem. Thus, the temporal oscillations of M associated with coherent tunneling do not actually take place. On the contrary, the time-scale of NVT in AF rings is microscopic, and much shorter than the decoherence time associated with spin–nuclei or spin–phonon interactions. This time-scale is just in the typical range accessed by cold neutrons. Moreover, INS is the technique of choice to study NVT since the neutron cross-section directly probes the autocorrelation function of n. This allows one to determine to which extent the autocorrelation is monochromatic (as in the NVT picture discussed above), and to measure the value of the tunnelling frequency.
4.2. Tunneling of the Néel vector An interesting aspect of the dynamics of AF rings is the possible occurrence of tunneling of the direction of the Néel vector P n ¼ Ni¼1 ð1Þi si =N (NVT) [11,38,39]. In the NVT regime, n would tunnel coherently between the two degenerate ±z directions. The two lowest energy states of the system would be even pffiffiffi and odd combinations of the classical Néel states (i.e. 1= 2ðj"; #; "; . . .i j#; "; #; . . .i)), and the T ¼ 0 time correlation functions, which has the general form
Fig. 4. Inelastic neutron scattering spectra for Fe7Mn collected at different temperatures. Solid lines are spectra calculated as described in the text. From Ref. [40].
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References
Fig. 5. Magnetic field dependence of the spectral weights of the zero-temperature autocorrelation function describing the Néel vector dynamics in Fe7Mn. The black line is the weight of the fundamental component, j h1 j nz j 0ij2 , the red one is the sum of the higher harmonics spectral weights. Calculations are based on the Hamiltonian (9), with parameters deduced from INS measurements. A Zeeman term corresponding to a magnetic field in the ring plane has been added to Eq. (9). Broken lines represent the results of calculations with S-mixing forced to zero. From Ref. [40].
A recent investigation of this phenomenon has been performed on the Fe7Mn ring, containing Fe3+ ions ðs ¼ 5=2Þ and one Mn2+ ion in the high-spin s ¼ 5=2 state [40]. The total spin of the ring in the ground state is therefore S ¼ 0. INS spectra collected at different temperatures on a polycrystalline sample are shown in Fig. 4. The data have been interpreted using Hamiltonian (9), with the following site-independent parameters: J n:n: ¼ 1:42 meV (nearestneighbor exchange), d ¼ 15 leV, j e=d j¼ 0:1 (ZFS), and a small next-nearest neighbor isotropic exchange contribution J n:n:n: ¼ 20 leV. Using these parameters, the magnetic-field dependence of the autocorrelation function has been calculated. The results are shown in Fig. 5, where the spectral weight j h1 j nz j 0ij2 is compared with the sum of all other components. Although the oscillations are almost monochromatic for B ¼ 0 and for values of B intermediate between two ground-state crossing fields (where the black line has maxima), the NVT regime is not achieved since j h1 j nz j 0ij2 is less than half the expected s2 value. It is worth to note that the fluctuation of the total spin plays an important role also in the Néel vector dynamics. In fact, if S mixing is artificially suppressed, the autocorrelation of nz is not characterized by a single frequency for B 6¼ 0 (see Fig. 5). 5. Conclusions The results discussed in this short review represent but a small fraction of the large number of INS studies of molecular nanomagnets. This extensive amount of work has been stimulated by the important discoveries of the group of Dante Gatteschi in Florence, which proved to be exceedingly fertile in fundamental research and promise to open new frontiers in technology. Following our first study on the Fe8 cluster in collaboration with Dante’s group, we have enjoyed a fruitful and pleasant collaboration over these years. We are deeply indebted to him for the excellent opportunities he has offered us.
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